Answer
(a)
Center: $C(-5,1)$
Vertices: $V_1(-9,1)$ and $V_2(-1,1)$
Foci: $F_1(-5-2\sqrt {3},1)$ and $F_2(-5+2\sqrt {3},1)$
(b)
Length of the major axis:
$2a=8$
Length of the minor axis:
$2b=4$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is horizontal):
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$\frac{(x+5)^2}{16}+\frac{(y-1)^2}{4}=1$
$\frac{[x-(-5)]^2}{4^2}+\frac{(y-1)^2}{2^2}=1$
$h=-5$
$k=1$
Center: $C(-5,1)$
$a=4$
$b=2$
$c^2=a^2-b^2=16-4=12$
$c=2\sqrt {3}$
The given equation can be obtained by shifting
$\frac{x^2}{4^2}+\frac{y^2}{1^2}=1$
left 5 units and upward 1 unit. In this equation:
Vertices: $V(±a,0)$
$V_1(-4,0)$ and $V_2(4,0)$
Foci: $F(±c,0)$
$F_1(-2\sqrt {3},0)$ and $F_2(2\sqrt {3},0)$
Now, shift these points 5 units to the left and 1 units upward:
Vertices: $V_1(-9,1)$ and $V_2(-1,1)$
Foci: $F_1(-5-2\sqrt {3},1)$ and $F_2(-5+2\sqrt {3},1)$
Length of the major axis:
$2a=8$
Length of the minor axis:
$2b=4$
